Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces

Pickrell, Doug; Xia, Eugene Z.

doi:10.1007/s00014-002-8343-1

Cited by 29 publications

(36 citation statements)

References 7 publications

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“…The cases of n = −1 (closed surface) and n = 0 are both direct consequences of [PX,Theorem (3.1)]. Hence for the rest of the paper, we assume n > 0.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary

Pickrell

Xia

2003

Transformation Groups

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“…The cases of n = −1 (closed surface) and n = 0 are both direct consequences of [PX,Theorem (3.1)]. Hence for the rest of the paper, we assume n > 0.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…By [PX,Remark (2.1.5)], the images of the maps S and S have full measure in K . Since the images of S and S are compact, S and S are actually onto maps.…”

Section: Ergodicitymentioning

confidence: 99%

Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary

Pickrell

Xia

2003

Transformation Groups

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“…Let M C (K) be its K-character variety, with the symplectic structure Ω as defined in [Gol84,Gol97], where C is a collection of n conjugation classes. Then Γ, the mapping class group of Σ, acts on M C (K) preserving Ω. Pickrell-Xia [PX02,PX03] established Γergodicity of the symplectic measure µ of Ω for g > 1 or n > 2 with g > 0. This was previously proved by Goldman [Gol97] when the simple factors of K are locally isomorphic to SU(2).…”

Section: Introductionmentioning

confidence: 99%

The mapping class group acts reducibly on 𝑆𝑈(𝑛)-character varieties

Goldman¹

2006

Primes and Knots

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“…Unlike the many cases in which Mod(Σ) acts properly discussed above, when G is compact, this measure-preserving action is ergodic on each connected component (Goldman [78], Pickrell-Xia [139], ). When G is noncompact, invariant open subsets of the deformation space exist where the action is proper (such as the subset of Anosov representations), but in general Mod(Σ) can act properly on open subsets containing non-discrete representations, even for PSL(2, R) ( [81,91,156]).…”

Section: Complex Projective 1-manifolds Flat Conformal Structures Anmentioning

confidence: 99%

Locally Homogeneous Geometric Manifolds

Goldman

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract.Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.Mathematics Subject Classification (2000). Primary 57M50; Secondary 57N16.Keywords. connection, curvature, fiber bundle, homogeneous space, Thurston geometrization of 3-manifolds, uniformization, crystallographic group, discrete group, proper action, Lie group, fundamental group, holonomy, completeness, development, geodesic, symplectic structure, Teichmüller space, Fricke space, hypebolic structure, Riemannian metric, Riemann surface, affine structure, projective structure, conformal structure, spherical CR structure, complex hyperbolic structure, deformation space, mapping class group, ergodic action. Historical backgroundWhile geometry involves quantitative measurements and rigid metric relations, topology deals with the loose quantitative organization of points. Felix Klein proposed in his 1872 Erlangen Program that the classical geometries be considered as the properties of a space invariant under a transitive Lie group action. Therefore one may ask which topologies support a system of local coordinates modeled on a fixed homogeneous space X = G/H such that on overlapping coordinate patches, the coordinate changes are locally restrictions of transformations from G. W. GoldmanIn this generality this question was first asked by Charles Ehresmann [55] at the conference "Quelques questions de Geométrie et de Topologie," in Geneva in 1935. Forty years later, the subject of such locally homogeneous geometric structures experienced a resurgence when W. Thurston placed his 3-dimensional geometrization program [158] in the context of locally homogeneous (Riemannian) structures. The rich diversity of geometries on homogeneous spaces brings in a wide range of techniques, and the field has thrived through their interaction.Before Ehresmann, the subject may be traced to several independent threads in the 19th century:• The theory of monodromy of Schwarzian differential equ...

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Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces

Abstract: Abstract. We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group.

Cited by 29 publications

References 7 publications

Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary

Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary

The mapping class group acts reducibly on 𝑆𝑈(𝑛)-character varieties

Locally Homogeneous Geometric Manifolds

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